Cavendish experiment
The Cavendish experiment, done in 1797–98 by British scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory,Boys 1894 p.355 and the first to yield accurate values for the gravitational constant and the mass of the Earth.Encyclopaedia Britannica 1910 p.385 'The aim experiments like Cavendish's may be regarded either as the determination of the mass of the Earth,...conveniently expressed...as its "mean density", or as the determination of the "gravitation constant", G'. Cavendish's experiment is generally described today as a measurement of G (Clotfelter 1987 p.210).Many sources state erroneously that this was the first measurement of G (or the Earth's density), such as – Scholar search}}. There were previous measurements, chiefly Bouguer (1740) and Maskelyne (1774), but they were very inaccurate (Poynting 1894)(Encyclopedia Britannica 1910). However, these were derived by others from Cavendish's result, which was a value for the Earth's density.Clotfelter 1987, p.210 The experiment was devised sometime before 1783McCormmach & Jungnickel 1996, p.336: A 1783 letter from Cavendish to Michell contains '...the earliest mention of weighing the world'. Not clear whether 'earliest mention' refers to Cavendish or Michell. by geologist John Michell,Cavendish 1798, p.59 Cavendish gives full credit to Michell for devising the experiment who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work, and after his death the apparatus passed to Francis John Hyde Wollaston and then to Henry Cavendish, who rebuilt the apparatus but kept close to Michell's original plan. Cavendish then carried out a series of measurements with the equipment, and reported his results in the Philosophical Transactions of the Royal Society in 1798.Cavendish, H. 'Experiments to determine the Density of the Earth', Philosophical Transactions of the Royal Society of London, (part II) 88 p.469-526 (21 June 1798), reprinted in Cavendish 1798 The experiment The apparatus constructed by Cavendish was a torsion balance made of a six-foot (1.8 m) wooden rod suspended from a wire, with a diameter lead sphere attached to each end. Two lead balls were located near the smaller balls, about away, and held in place with a separate suspension system.Cavendish 1798, p.59 The experiment measured the faint gravitational attraction between the small balls and the larger ones. The two large balls were positioned on alternate sides of the horizontal wooden arm of the balance. Their mutual attraction to the small balls caused the arm to rotate, twisting the wire supporting the arm. The arm stopped rotating when it reached an angle where the twisting force of the wire balanced the combined gravitational force of attraction between the large and small lead spheres. By measuring the angle of the rod, and knowing the twisting force (torque) of the wire for a given angle, Cavendish was able to determine the force between the pairs of masses. Since the gravitational force of the Earth on the small ball could be measured directly by weighing it, the ratio of the two forces allowed the density of the earth to be calculated, using Newton's law of gravitation. Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water (due to a simple arithmetic error, found in 1821 by F. Baily, the erroneous value 5.48 ± 0.038 appears in his paper).Poynting 1894, p.45 To find the wire's torsion coefficient, the torque exerted by the wire for a given angle of twist, Cavendish timed the natural oscillation period of the balance rod as it rotated slowly clockwise and counterclockwise against the twisting of the wire. The period was about 7 minutes. The torsion coefficient could be calculated from this and the mass and dimensions of the balance. Actually, the rod was never at rest; Cavendish had to measure the deflection angle of the rod while it was oscillating.Cavendish 1798, p.64 Cavendish's equipment was remarkably sensitive for its time. The force involved in twisting the torsion balance was very small, 1.47 x 10–7 N,Boys 1894 p.357 about 1/50,000,000 of the weight of the small ballsCavendish 1798 p. 60 or roughly the weight of a large grain of sand.A 2 mm sand grain weighs ~13 mg. To prevent air currents and temperature changes from interfering with the measurements, Cavendish placed the entire apparatus in a wooden box about thick, tall, and wide, all in a closed shed on his estate. Through two holes in the walls of the shed, Cavendish used telescopes to observe the movement of the torsion balance's horizontal rod. The motion of the rod was only about .Cavendish 1798, p. 99, Result table, (scale graduations = 1/20 in ≈ 1.3 mm) The total deflection shown in most trials was twice this since he compared the deflection with large balls on opposite sides of the balance beam. Cavendish was able to measure this small deflection to an accuracy of better than one hundredth of an inch using vernier scales on the ends of the rod.Cavendish 1798, p.63 Cavendish's experiment was repeated by Reich (1838), Baily (1843), Cornu & Baille (1878), and many others. Its accuracy was not exceeded for 97 years, until C. V. Boys' 1895 experiment. In time, Michell's torsion balance became the dominant technique for measuring the [[gravitational constant|gravitational constant (G'')]], and most contemporary measurements still use variations of it. This is why Cavendish's experiment became ''the Cavendish experiment.McCormmach & Jungnickel 1996, p.341 Did Cavendish determine G''? It is not unusual to find books that state erroneously that Cavendish's purpose was determining the [[Gravitational constant|gravitational constant (''G)]], 'The apparatus used in 1798 by Henry Cavendish to measure the gravitational constant' – Scholar search}} 'Cavendish claimed he was weighing the Earth, but what he was measuring was the coefficient G...' – Scholar search}} 'Cavendish was able to measure the force, the two masses, and the distance, and thus determine the gravitational constant G' . 'torsion balance was...modified by Cavendish to measure G.' 'Cavendish calculates the gravitational constant, which in turn gives him the mass of the earth...' and this mistake has been pointed out by several authors.Clotfelter 1987McCormmach & Jungnickel 1996, p.337Hodges 1999Lally 1999 In actuality, Cavendish's only goal was to measure the density of the Earth; he called it 'weighing the world'. The method Cavendish used to calculate the Earth's density consists in measuring the force on a small ball caused by a large ball of known mass, and comparing it with the force on the small ball caused by the Earth, so the Earth can be calculated to be N'' times more massive than the large ball without the need to obtain a numeric value for ''G. The gravitational constant does not appear in Cavendish's paper, and there is no indication that he regarded it as a goal of his experiment. One of the first references to G'' is in 1873, 75 years after Cavendish's work.Cornu, A. and Baille, J. B. (1873), Mutual determination of the constant of attraction and the mean density of the earth, ''C. R. Acad. Sci., Paris Vol. 76, 954-958. In Cavendish's time, G'' did not have the importance among scientists that it has today; it was simply a proportionality constant in Newton's law.Boys 1894, p.330 In this lecture before the Royal Society, Boys introduces G and argues for its acceptance The purpose of measuring the force of gravity was instead to determine the Earth's density. This was a much-desired quantity in 18th-century astronomy, since once the Earth's density was known, the densities of the Moon, Sun, and the other planets could be found from it.Poynting 1894, p.4 There had been previous attempts to measure the density of the Earth such as the Schiehallion experiment by the Royal Society in 1774. A further complication is that up through the mid-nineteenth century, scientists did not use a specific unit of measurement for force. This unnecessarily tied ''G to the mass of the Earth, as opposed to G'' being recognized as a universal constant. However, even though Cavendish did not report a value for ''G, the results of his experiment allowed it to be determined. During the late 1800s, as scientists began to recognize G'' as a fundamental constant of nature, they calculated it from Cavendish's accurate results, thus:MacKenzie 1900, p.vi : G = g\frac{R_\text{earth}^2}{M_\text{earth}} = \frac{3g}{4\pi R_\text{earth}\rho_\text{earth}}\, After converting to SI units, Cavendish's value for the Earth's density, 5.448 g cm−3, gives :''G = 6.74 × 10−11 m3 kg−1 s−2, which differs by less than 1% from the currently accepted value: 6.67259 × 10−11 m3 kg−1 s−2. Derivation of G and the Earth's mass For the definitions of terms, see the drawing below and the table at the end of this section. The following is not the method Cavendish used, but shows how modern physicists would use his results.Cavendish Experiment, Harvard Lecture Demonstrations, Harvard Univ.Poynting 1894, p.41Clotfelter 1987 p.212 explains Cavendish's original method of calculation From Hooke's law, the torque on the torsion wire is proportional to the deflection '' \theta'' of the balance. The torque is '' \kappa\theta'' where '' \kappa'' is the torsion coefficient. However, the torque can also be written as a product of the attractive forces and the distance to the wire. Since there are two pairs of balls, each experiencing force F'' at a distance ''L / 2 from the axis of the balance, the torque is LF. Equating the two formulas for torque gives the following: : \kappa\theta\ = LF \, For F'', Newton's law of universal gravitation is used to express the attractive force between the large and small balls: : F = \frac{G m M}{r^2}\, Substituting F into the first equation above gives : \kappa\theta\ = L\frac{GmM}{r^2} \qquad\qquad\qquad(1)\, To find the torsion coefficient ( \kappa\, ) of the wire, Cavendish measured the natural resonant oscillation period ''T of the torsion balance: : T = 2\pi\sqrt{I/\kappa} Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls: : I = m(L/2)^2 + m(L/2)^2 = 2m(L/2)^2 = mL^2/2\, , and so: : T = 2\pi\sqrt{\frac{mL^2}{2\kappa}}\, Solving this for '' \kappa'', substituting into (1), and rearranging for G'', the result is: : G = \frac{2 \pi^2 L r^2}{M T^2} \theta\, Once ''G has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the Earth's mass and density: : mg = \frac{GmM_{earth}}{R_{earth}^2}\, : M_{earth} = \frac{gR_{earth}^2}{G}\, : \rho_{earth} = \frac{M_{earth}}{4 \pi R_{earth}^3/3} = \frac{3g}{4 \pi R_{earth} G}\, See also * Schiehallion experiment References * * Online copy of Cavendish's 1798 paper, and other early measurements of gravitational constant. * Establishes that Cavendish didn't determine G. * * * Discusses Michell's contributions, and whether Cavendish determined G. * * * Review of gravity measurements since 1740. * Notes External links * [http://www.sas.org/tcs/weeklyIssues_2005/2005-07-01/feature1/index.html Sideways Gravity in the Basement, The Citizen Scientist, July 1, 2005], retrieved Aug. 9, 2007. Homebrew Cavendish experiment, showing calculation of results and precautions necessary to eliminate wind and electrostatic errors. * Measuring Big G, Physics Central, retrieved Aug. 9, 2007. Recent experiment at Univ. of Washington to measure the gravitational constant using variation of Cavendish method. * The Controversy over Newton's Gravitational Constant, Eot-Wash Group, Univ. of Washington, retrieved Aug. 9, 2007. Discusses current state of measurements of G. * Model of Cavendish's torsion balance, retrieved Aug. 28, 2007, at Science Museum, London. * Weighing the Earth - background and experiment Category:Physics experiments Category:1790s in science de:Gravitationswaage es:Experimento de Cavendish fa:آزمایش کاوندیش fr:Balance de torsion he:ניסוי קוונדיש kk:Кавендиш тәжірибесі ja:キャヴェンディッシュの実験 pl:Waga skręceń pt:Experiência de Cavendish ru:Эксперимент Кавендиша sr:Кевендишов оглед vi:Thí nghiệm Cavendish